Örebro University
Örebro University School of Business
Masters in Applied Statistics
Sune karlsson
Farrukh Javed
JUNE, 2016
Modeling and Forecasting Unemployment Rate
In Sweden using various Econometric Measures
Acknowledgments
First, and foremost, I would like to thank the almighty God for giving me the opportunity to pursue my graduate study at Department of Applied Statistics, Orebro University.
I owe the deepest gratitude to Prof.Sune karlsson, my thesis advisor for his valuable and constructive comments and encouragements throughout my study.
My deepest thanks go to my family, who have supported me all the way to fulfill my dream.
TABLE OF CONTENTS
ACRONYMS………...………..I
ABSTRACT………...………...II
CHAPTER1:INTRODUCTION...1
1.1Introduction ………..………...1
1.2 Objectives of the Study...4
CHAPTER 2: LITERATURE REVIEW...5
2.1 Review of Literature…...………...………...5
CHAPTER 3 : DATA AND METHODOLOGY...…...7
3.1 Data...……….………...7
3.2 Methodology…………...……….………...7
3.3 Stationary Test...8
3.4 Lag length selection...11
3.5 Time series models………...11
3.5.1 Univariate time series model...12
3.5.1.1 Seasonal Autoregressive Integrated Moving Average Model (SARIMA)...12
3.5.1.2 Self-Exciting Threshold Autoregressive (SETAR) Model...14
3.5.2.Multivariate time series model...17
3.5.2.1. Vector Autoregressive Model ...17
3.6 Model checking.………...21
3.6.1 Residual Analysis...22
3.6.1.1 Residual autocorrelation test...22
3.6.1.2 Residuals normality test...24
3.7 Forecasting ... 26
3.7.1 Out sample forecasting method...27
CHAPTER 4: RESULT AND DISCUSSION………...30
4.1 Descriptive Analysis………...30
4.2 Stationary test Analysis.………...30
4.3 Modeling Seasonal Autoregressive integrated moving average model...………...35
4.4 Modeling Self Exciting Threshold Autoregressive Model………...47
4.5 Modeling of Vector Autoregressive Model...40
4.6 Comparison of Models Forecasting Performance...44
4.7 Forecasting Result...45
CHAPTER 5: CONCLUSION AND RECOMMENDATION………...47
5.1 Conclusions.…...………...47
5.2 Recommendations………...………...47
REFERENCES………...48
ANNEX1: STATA and R outputs...………...…...50
Table A1: SARIMA Models Comparison………...………...50
Table A2: Result for TAR model...……….………...………...51
Table A3: VAR order selection test using lag four, six and seven...………...52
Table A4: Pair-wise Granger-causality tests……….…...………...53
ACRONYMS
VAR Vector Autoregressive
SETAR Self Exciting threshold autoregressive
SARIMA Seasonal autoregressive integrated moving average AIC Akaike Information Criteria
BIC Schwartz and Bayes Information Criterion HQ Hannan-Quin
OECD Organization for Economic Cooperation and Development ADF Augmented Dickey-Fuller
PP Phillips -Perron LM Lagrange multiplier
ACF Autocorrelation Function
PACF Partial Autocorrelation Function RMSE Root mean square error
MAE Mean absolute error
MAPE Mean Absolute percentage error
DM Diebold-Mariano
ABSTRACT
Unemployment is one of the several socio-economic problems exist in all countries of the world. It affects people's living standard and nations socio-economic status. The main objective of this study is modeling and forecasting unemployment rate in Sweden. The study exploits modeling unemployment rate using SARIMA, SETAR, and VAR time series models determine the goodness of fit as well as the validity of the assumptions and selecting an appropriate and more parsimonious model thereby proffer useful suggestions and recommendations. The fit of models was illustrated using 1983-2010 of unemployment rate quarterly data obtained from OECD. The study provided some graphical and numerical methods for checking models' adequacy. The tested models are well fit and adequate based on the assumptions of the goodness of fit. Moreover, using different stationary test, some variables proved to be integrated of order one. The Granger causality test shows the causality between unemployment rate, GDP percentage change of previous period, and industrial production but inflation rate does not have causality relation with all variables. Besides, Johansen cointegration test of cointegrating vectors in the variables shows no cointegration found. The out-of-sample forecasting performance evaluation is performed using data from 2011-2015 with recursive method. Findings have shown that both the seasonal autoregressive integrated moving average and self-exciting threshold autoregressive models outperform the VAR model and have the same forecasting performance in both in-sample and recursive out-of-in-sample forecasting performance. The eight quarter forecasted values from both models have small difference while all values are placing within the 95% forecasting confidence interval of SARIMA model. The finding of the study further indicated that short-term forecasting is better than long term. As short-term forecasting is better, there should be a continuous investigation of appropriate models which used to predict the future values of the unemployment rate.
Keywords: Unemployment rate, Modeling, Forecasting, Out sample forecasting.
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CHAPTER ONE
1.1 INTRODUCTION
Unemployment is one of the several socio-economic problems exist in all countries of the world. It affects people's living standard and nations' socio-economic status.
The main reason for increasing unemployment rate is the deficiency of demand in the economy to maintain full employment. When there is less demand, companies need less labor input, leading them to cut hours of work or laying people off. Though unemployment is mainly caused by a fundamental shift in an economy, its frictional, structural, and cyclical behavior also contributes to its existence.
Frictional unemployment is unemployment which exists in any economy due to the inevitable time delays in finding new employment in a free market. Structural unemployment occurs for many reasons, such as people may lack the needed job skills or they may live far from locations where jobs are available but unable to move there. Besides, sometimes people may be unwilling to work because existing wage levels are too low. Consequently, while jobs are available, there is a serious mismatch between what companies need and what workers can offer. In general, structural unemployment is increased by external factors like technology, competition, and government policy. Moreover, cyclical unemployment is a factor for unemployment related with the cyclical trend of economic indicators which exist in a business cycle. Cyclical unemployment declines at business cycles increased in output since the economy gets maximized. On the other hand, when the economy measured by the gross domestic product (GDP) declines, the business cycle becomes lower and cyclical unemployment gets a rise. Economists define cyclical unemployment as the result of companies have a lack of demand for labor to hire individuals, who are looking for work. It is logical when the economy gets down the lack of employer demand will exist.
At any time and economy status of a country, there is some level of unemployment. Due to its fractional and structural behavior, unemployment is positive rather zero. The natural level of unemployment is the unemployment rate when an economy is operating at full capacity. The
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quantity of labor supplied equals the quantity of labor demanded at the natural level. Mainly, the natural rate of unemployment is determined by an economy's production possibilities and economic institutions. Besides, it is a rate at which there is no tendency for inflation to accelerate or decelerate. Consequently, when an economy is at the natural rate inflation is constant. For this reason, the natural rate of unemployment is sometimes called constant inflation rate of unemployment.
THE SOCIO ECONOMIC EFFECT OF UNEMPLOYMENT
1). The social effects of unemploymentUnemployment affects both the individuals and their families. In the long run, it also affects the society. Unemployment causes for a mental health problem and on the physical well-being of individuals. Hammarstrom and Janlert (1997), stated that those who are unemployed expected to experience different emotions such as sadness, hopelessness, humiliation, worry, and pain. Besides, according to Britt, 1994; Weich and Lewis, 1998; Reynolds, 2000, report different crimes prevail in a society when there is a high unemployed group in the population. Health problems, drug abuse, and similar problems are highly associated with unemployment.
2). Unemployment affects economy
Economists describe unemployment is a lagging indicator of the economy, as the economy usually recovers before the unemployment rate starts to rise again. However, unemployment causes a sort of wave effect across the economy. As people losing their job, they do not pay state and federal income taxes, and additional sales tax revenue. Instead as a laid off worker, they could immediately cut back to their unnecessary cost due to less disposable income and this cause for less money to be spent in the economy, driving to more people to miss their jobs. Consequently, the process recycles unless it is broken by developing a policy and plan.
In general, the rising of unemployment rate highly affects different economy factors such as personal income, cost of health, health care quality, leaving standard and poverty. All those affect the entire systems of the economy as well as the society.
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Trend of Sweden unemployment rate
In Sweden, the unemployment rate is defined as the number of unemployed individuals calculated as a percentage of individuals in the labor force, which includes both the employed and unemployed.
The average unemployment rate in Sweden from 1980 until 2016 is 5.88 percent. During this period, the highest was 10.50 percent reported in June 1997 and the lowest was 1.30 percent reported in July 1989. The following graph shows the pattern of the unemployment rate in Sweden for last six consecutive years.
Figure: unemployment rate of sweden for six consecutive years.
The above graph reveals that the rate of unemployment in 2010 is 8.6%, increased from 8.3% a year earlier, in 2015 the rate is 7.4%, decline from 7.9% in the previous year. Besides, no matter the rate is increased or decreased it varied between 0.1 and 2.1 percent in each year.
While unemployment rate decreases compared to its previous value, it sometimes higher than market expectations. For instance, the rate of unemployment in February 2016 is 7.6% lower than a year earlier 8.4% but higher than the market expectations 7.4%, Economy trade (2016).
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Since unemployment is closely related to the state of the economy predicting the unemployment rate has value to various economic decisions. It is used for both monetary policy makers to serves as an indicator of the stance of the macroeconomic in general and carries information regarding inflationary pressure as well as for fiscal policy makers related to the government expenditure and income due to its relationship with, for instance, income taxes and unemployment benefits. Therefore, signals of future unemployment rates are necessary for policy and decision makers to plan and strategize before time. Consequently, this study focuses on modeling and forecasting the rate of unemployment in Sweden.
1.2 General objective of the study
The main objective of the study is to modeling and forecasting the unemployment in Sweden. In this case, the study fit univariate and multivariate time series models for the unemployment rate and other macroeconomic variables which can be used to achieve the objective.
Specific objectives
Modeling the linear and asymmetry behavior of unemployment rate
Comparing the forecasting performance of different models
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CHAPTER TWO
2.1 LITERATURE REVIEW
Modeling and forecasting of macroeconomic variables used to address different issues related to the economic state of the countries. Researchers have used various time series models for modeling and forecasting of macroeconomic variables. The unemployment rate is one of the macroeconomic variables, modeling, and forecasting of it have a great importance for many economic decisions. Different models have been used to modeling and forecasting as well as to compare the forecasting performance of models for the unemployment rate of several countries. Like any other economic variables modeling of unemployment rates have been analyzed by building econometric models, often related to stationary time series, seasonality and trend analysis, and exponential smoothening to the simple OLS technique including autoregressive integrated moving average(ARIMA) models.
The suitability of the ARIMA models for forecasting macroeconomic variables is studied by various researchers; Power and Gasser (2012) investigate that an ARIMA (1,1,0) model has better forecasting performance for unemployment rates in Canada. Besides, an ARIMA (1,2,1) model is suitable for forecasting the unemployment rate in Nigeria, as reported by Etuk et al.(2012).
VAR model is one of the most useful time series models to describe the dynamic behavior of macroeconomic variables and to forecast. Clements and Hendry (2003) stated that the accuracy of forecasts based on VAR models can be measured using the trace of the mean-squared forecasts error matrix or generalized forecasts error second moment. Robinson (1998) demonstrated better accuracy for predictions of some macroeconomic variables based on VAR models compared to other models, like transfer functions. Finally, Lack (2006) found that combined forecasts based on VAR models are a good strategy for improving predictions’ accuracy.
Kishor and Koenig (2012) made predictions for macroeconomic variables like unemployment rate using VAR models and taking into account that data are subject to revisions.
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The unemployment rate reacts in a different way to contraction and expansion phases of the general business cycle. It faster moves upward in a general business slowdown and slowly downward in speedup phases. The asymmetric nature of the business cycle can be considered the main source of nonlinearity in the unemployment time series. The classical linear models are not able to describe these dynamic asymmetries and nonlinear time series models would be required. However, the asymmetric behavior of unemployment rate can be modelled using nonlinear time series model. Skalin and Terasvirta (1998) propose STAR model in order to capture the asymmetry property of unemployment rate. They assume that unemployment rate is a stationary nonlinear variable.
Peel and Speight (2000), examine whether a Self-Exciting Threshold Autoregressive (SETAR|) models are able to provide better out-of-sample forecasts compared to an Autoregressive model using Uk unemployment sample data from February 1971 to September 1991. The result shows that SETAR models have better forecasting performance relative to AR models in terms of RMSE. Koop and Potter (1999) use threshold autoregressive (TAR) for modeling and forecasting the US monthly unemployment rate. Rothman (1998) compares out-of-sample forecasting accuracy six nonlinear models, and Parker and Rothman (1998) model the quarterly adjusted rate with AR(2) model. Proietti (2001) used seven forecasting models (linear and non-linear) to examine the out-of-sample forecasting for the US monthly unemployment rate. The result reveals that linear models have better forecasting performance than nonlinear models.
Besides, Jones(1999), Gil-Alana(2001) reported, in different research papers, the modeling and forecasting of the unemployment rate in the UK. Johns (1999) examines the forecasting comparison between AR(4), AR(4)-GARCH(1,1), SETAR(3,4,4), Neural network and Naïve forecast of UK monthly unemployment rate with the sample data from January 1960 to August 1996. The result reveals that SETAR model is better than the others for short period forecasts, while non-linearity was present in the data.
Moreover, Gil-Alana (2001) used a Bloomfield exponential spectral model for modeling UK unemployment rate, as an alternative to the ARMA models. The results reveal that this model is reasonable to model UK unemployment rate.
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CHAPTER THREE
3. DATA AND METHODOLOGY
3.1. Data:This study used Sweden unemployment rate quarterly data from the first quarter of 1983 to the fourth quarter of 2015, with a total number of 132 observations collected from OECD. The first 112 observations are used to model estimation and the rest 20 observations to evaluate model forecasting performance.
Variables in the study
Since the study uses both univariate and multivariate time series models to model the unemployment rate in Sweden, the study considers some additional economic variables which have a direct or indirect relation with unemployment. According to economic theory, there is a direct relationship between gross domestic product and unemployment rate. The GDP get lower when unemployment rate becomes above its natural rate and vice-versa. There is also a relationship between unemployment rate and industrial production, potential output measures the productive capacity of the economy when unemployment is at its natural rate. In most cases the produced output is proportional to the level of the inputs (capital and labor). Thus, an increasing unemployment above its natural rate is related to the falling of output below its potential and vice-versa. Moreover, there is an indirect relation between the rate of unemployment and the rate of inflation. Consequently, in addition to model unemployment rate alone, the study also model it together with GDP percentage change of previous period, industrial production, and inflation rate.
3.2. METHODOLOGY
Time series can be defined as any series of measurements taken at different times, can be divided into univariate and multivariate time series. Univariate time series analysis uses one series. However, the multivariate time series analysis involves more than one series data sets used when one wants to model and explain the effect and relation among time series variables.
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3.3. TESTING STATIONARY
Unit root test without structural break
Before fitting a particular model to time series data, the stationarity of a series must be checked. Stationarity occurs in a time series when the mean and autocovariance of the series remains constant over the time series. It means that the joint statistical distribution of any collection of the time series variates never depends on time. Therefore, the stochastic process yt is said to be
stationary if:
i. E(yt),constant for all value of t ...(1)
ii. The Cov(yt,ytj)j E
(yt )(ytj)T
Tj for all t and j=0,1,2, ...(2)Equation (1) means that yt have the same finite mean μ through the process and (2) requires that
the autocovariance of the process do not depend on t but just on the time period j, the two vectors yt and yt-j are apart. Therefore, a process is stationary if its first and second moments are
time invariant.
Usually, differencing may be needed to achieve stationarity. Several methods have been developed to test the stationarity of a series. The most common ones are Augmented Dickey-Fuller (ADF) test due to Dickey and Dickey-Fuller (1979, 1981), and the Phillip-Perron (PP) due to Phillips (1987) and Phillips and Perron (1988). The following discussion outlines the basics features of unit root tests (Hamilton, 1994).
Consider a simple AR (1) process:
yt yt1t...(3)
Where yt is the variable of interest, t is time index, is parameter to be estimated, and t is assumed to be a white noise. If 1, yt is a non stationary series and the variance of yt
increases with time and approaches infinity. If 1, yt is a stationary series. Thus, the
hypothesis of (trend) stationarity can be evaluated by testing whether the absolute value of ρ is strictly less than one. The test hypothesis is:
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H0: The series is not stationary (ρ=1) H1: The series is stationary (ρ<1)
3.3.1. Augmented Dickey-Fuller (ADF) Test
The standard Dickey-Fuller test is conducted by estimating equation (3) after subtracting yt1 from both side of the equation
yt yt1t...(4)
where ᾳ=ρ-1 and yt ytyt1. The null and alternative hypothesis may be written as, H0: 0
H0 : 0...(5)
and evaluated using the conventional t-ratio for ᾳ:
/( ( )) se t ...(6) Where
is the estimate of , and se(
) is the coefficient standard error.
Under the null hypothesis of the unit root test the DF test statistics does not follow the conventional student t-distribution instead asymptotic t-distribution.
The simple Dickey-Fuller unit root test described above is valid only when the series is an AR(1) process. If the series is correlated at higher order lags, the assumption of white noise disturbances ԑt is violated. The Augmented Dickey-Fuller (ADF) test constructs a parametric
correction for higher-order correlation by assuming that the series follows an AR(p) process and adding lagged difference terms of the dependent variable y to the right-hand side of the test regression:
yt yt11yt12yt2...pytpUt...(7)
This augmented specification is then used to test (5) using the t-ratio (6). An important result obtained by Fuller is that the asymptotic distribution of the t-ratio for ᾳ is independent of the number of lagged first difference included in the ADF regression. Moreover, while the
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assumption thatyt follows an autoregressive (AR) process may seem restrictive, said and Dickey (1984) demonstrate that the ADF test is asymptotically valid in the presence of a moving average (MA) component, provided that sufficient lagged difference terms are included in the test regression.
3.3.2. The Phillips -Perron (PP) Test
Phillips and Perron (1988) proposes an alternative (nonparametric) method of controlling for serial correlation when testing for a unit root. The PP method estimates the non-augmented DF test equation (5), and modifies the t-ratio of the ᾳ coefficient so that serial correlation does not affect the asymptotic distribution of the test statistics. The PP test is based on the statistic:
s f se f T f t t 1/2 0 0 0 2 / 1 0 0 2 ) ( ...(8) where is the estimate, t is the t-ratio of ᾳ, se(
) is coefficient standard error and s is the standard error of the test regression. In addition, 0is a consistent estimate of the error variance in (5) (calculated as (T-K)s2/T) calculated, where k is the number of regressors). The remaining term, f0, is an estimator of the residual spectrum at frequency zero.
Unit root test with structural break
3.3.3 Zivot and Andrews Test
Zivot and Andrews endogenous structural break test is a sequential test which uses the full sample and a different dummy variable for each possible break date. The break date is selected where the t-statistics of a unit root ADF test is at a minimum (most negative). Consequently, a break date will be chosen when the evidence does not support the null hypothesis of a unit root. Zivot and Andrews perform a unit root test with three conditions such as a structural break in the level of the series; a one-time change in the slope of the trend, and a structural break in the level and slope of the trend function of the series. Therefore, to test for the null of a unit root against the alternative of a stationary structural break, Zivot and Andrews (1992), use the following equations corresponding to the above three conditions.
11 t t k j j t t t c y t DU d y y
1 1 1 (condition one) t t k j j t t t c y t DT d y y
1 1 1 (condition two) t t k j j t t t t c y t DU DT d y y
1 1 1 (condition three)where DUt is an indicator variable for a mean shift obtained at each possible break-date (BD) while DTt is corresponding trend shift variable. Then,
otherwise 0 BD t if 1 t DU and otherwise 0 BD t if BD t DTt
The null hypothesis of the three models assumes α=0, which implies that the series yt contains a
unit root without any structural break, while the alternative assumes α< 0, suggest that the series is a trend-stationary process with a one-time break occurring at an unknown time point. The Zivot and Andrews method considers every point as a potential break-date (BD) and runs a regression for every possible break-date sequentially. Amongst all possible break-points (BD*), the procedure selects as its choice of break-date (BD) the date which minimizes the one-sided
t-statistic for testing ( 1)1
.
3.4. LAG LENGTH SELECTION
Choosing the lag length has strong implication for choosing models. Taking too few lags cause for misspecification of model correctly whereas taking too many lags could cause to increase the error in the forecasts. Therefore, taking optimal lag length is important.
3.5.TIMES SERIES MODELS
The study uses both linear and nonlinear univariate, and multivariate time series models. It has two parts, the first part performs modeling and estimation of the unemployment rate; the second part deals with forecasting of the rate of unemployment.
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3.5.1. Univariate Time Series Models
3.5.1.1 Seasonal Autoregressive Integrated Moving Average Model (SARIMA)
For unemployment rate time series, a seasonality might need to consider in ARIMA model. This process is known as a seasonal process and it drives ARIMA into SARIMA process. Seasonal Autoregressive Integrated Moving Average (SARIMA) model is a generalized form of ARIMA model which accounts for both seasonal and non-seasonal characterized data. Similar to the ARIMA model, the forecasting values are assumed to be a linear combination of past values and past errors. The SARIMA model also sometimes referred to as the Multiplicative Seasonal Autoregressive Integrated Moving Average model, is denoted as ARIMA(p,d,q) (P,D,Q)S. The corresponding lag form of the model is:
t s t D s d s L L y L L L L ( ) ( )(1 ) (1 ) ( ) ( )
Using L of order p and q respectively the model includes the following AR and MA characteristic polynomials: L L L p1Lp1pLp 2 2 1 ... 1 ) ( L L L q1Lq1qLq 2 2 1 ... 1 ) (
Also seasonal polynomial functions of order P and Q respectively as represented below: (Ls)11Ls2L2s ...p1L(p1)spL(p)s
(Ls)11Ls 2L2s...Q1L(Q1)sQL(Q)s
Where, ytis a time series
t- white noise error terms
p,d,q - the order of non-seasonal AR, differencing, and non-seasonal MA respectively. P,D,Q- the order of seasonal AR, differencing, and seasonal MA respectively.
S-seasonal order, in this case S=4 for quarterly data. Llag operator t t k
k
y y
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HEGY test of Seasonality
The first procedure towards designing of SARIMA model is to examine whether the series satisfy the stationarity condition. This study used Hylleberg Engle-Granger-Yoo (HEGY) test suggested by Hylleberg et al. (1990) to test for the presence of seasonal unit root in the observable series. HEGY test is a test for seasonal and non-seasonal unit root in a time series. A time series 𝑦𝑡 is considered as an integrated seasonal process if it has a seasonal unit root as well
as a peak at any seasonal frequency in its spectrum other than the zero frequency. The test is based on the following auxiliary regression:
4yst 1y1,t12y2,t13y3,t1t Where, 4 1L4 (1L)(1L)(1L2)
y1,t (1LL2L3)yt
y2,t (1LL2L3)yt
y3,t (1L2)yt
The null hypothesis for HEGY test is
0 : vs. 0 : 1 1 1 0 H
H , H0:2 0 vs. H1:2 0, and the joint
0
: 3 4
0
H vs. H1:3 0and/ or 4 0.
when 1,2 0, the null hypothesis of the presence of a unit root (non-seasonal unit root)
cannot be rejected. Besides, when π3 = π4 = 0, the hypothesis of presence of seasonal roots
cannot be rejected and they jointly tested using F-test which has a nonstandard distribution.
SARIMA Order Selection
The seasonal and non-seasonal autoregressive and moving average component lags p, P and q, Q is determined by plotting the ACF and PACF. The plot gives information about the internal correlation between time series observations at different times apart to provide an idea about the seasonal and non-seasonal lags. Both the ACF and PACF have spikes and cut off at lag k and lag ks at the non-seasonal and seasonal levels respectively. The order of the model is given by the
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Choosing best SARIMA model
Based on the plots of both autocorrelation and partial autocorrelation function there could be a different SARIMA(p,d,q)(P,D,Q) model with different significant lags of p,P and q,Q. Thus, a SARIMA model with an optimal lag length of seasonal and nonseasonal components should chosen using criteria. Hence, the study used both Akaike Information Criterion (AIC) and Schwartz and Bayes Information Criterion (BIC). The Akaike and Schwarz and Bayes information criterions are computed as follows:
AIC2logL2m BIClogLmlogL
Where m pPqQ is the number of parameters in the model and L is the likelihood function. A best SARIMA(p,d,q)(P,D,Q) model is the one with small AIC and BIC.
3.5.1.2 Self Exciting Threshold Autoregressive (SETAR) Model
Self-Exciting Threshold Autoregressive (SETAR) model is called a piecewise linear model or regime-switching model. It consists of k, AR(p) parts where one process change to another according to the value of an observed variable, the threshold. Once the series cross the threshold value, the process takes on another value. In a TAR model, AR models are estimated separately in two or more intervals of values as defined by the dependent variable. These AR models may or may not be of the same order. For convenience, it’s often assumed that they are of the same order.
Suppose a time series yt follows the threshold autoregressive model TAR(k;p,d):
( ), ( 1 ) 1 ) ( ) ( 0 j j n j t i t p i j i j t y r r r y
...(10)where rj are the threshold variables which belongs to r0 r1...rn ; (j1,...,n)and k is the number of regimes; t(j) ~iid(0,2). d is the threshold lag and p is the autoregressive
order. In this model, there are k autoregressive parts in each different regime which divided by k-1 thresholds rj. The series will have different behavior in different regime, while they follow the
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AR (p) model in each regime. When the threshold variable rj ytd with a delay parameter d, the dynamic of yt is determine by its own lagged value ytd the TAR model is called Self-exciting or SETAR.
The simple model of SETAR is a model with two regimes k=2, p-order, d-delay and one threshold value r. The model SETAR(2,p,d) presented as follows:
r y y r y y y d t i t p i i d t i t p i i t t t if , if , ) 2 ( 1 ) 2 ( ) 2 ( 0 ) 1 ( 1 ) 1 ( ) 1 ( 0 ...(11)Order selection of SETAR model
The selection procedure of an optimal lag order of SETAR model for each regime is the same with the selection of AR order in ARIMA model.
Testing for and estimation of the threshold
In order to determine the number of regimes, it is important to test whether the SETAR model of Equation (10) is statistically significant relative to a linear AR(p). The null hypothesis is:
H0:no nonlinear threshold
Tsay’s test of nonlinearity
The Tsay(1989) F-test uses an arranged autoregression with recursive least square estimation. Observations need to be sorted from the smallest observation to the largest.
Suppose a set of observations (yt,1,yt1,...,ytp), for t p1,...,n for AR(p) model. There are
two conditions for the threshold variable ytd. The first one is when d p1, the threshold variables are (yp1d,..., ynd) and the second one is, when d p1, then, the threshold variables
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,...,
(yh ynd , where hmax{1,p1d}. suppose k=2 and a threshold r1 the TAR (2; p, d) can be rewrote as y y d v d i s p v v d i i i
(1) if 1 ) 1 ( ) 1 ( 0 y d v d i s p v v i i
(2) if 1 ) 2 ( ) 2 ( 0 ...(12)Where i is the time index of the i
th
smallest observation of {yh,..., ynd}; and s satisfies
. 1 1 s s r y y
In two different regimes, there are two arranged autoregressions separated by threshold r1. Therefore, i=1,2,...,n-d-h+1 and n-d-h+1 is effective sample size.
Suppose the autoregressions start from b observations and since there are n-d-h+1 observations in arranged autoregression, there are n-d-b-h+1 predictive residuals. Therefore, the least squares regression is , for i b 1,...,n-d-h 1 , 0 2
d i v d i i y e p v i v d ...(13)From equation (13) it's possible to get least squares residual, 2 t e and 2 t
and compute the associated F-statistics: ) ( ) 1 ( ) , ( 2 2 2 h p b d n p e d p F t t t
...(14) Thus, F statistics approximately distributed to F-distribution with degree of freedom (p1)
and (ndbh). When there is threshold nonlinearity, F F(p1,ndbh)
and the null hypothesis will reject.
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Selecting the Delay d and Locating the Threshold Values
The parameter delay d is a set of possible positive integers and its value is related with order
p. The possible value d is less than order p.
Grid search method
The problem estimating parameters of SETAR model (0,...,p) is the unknown value of
the thresholds. However, when the threshold values are known, parameters in SETAR model are estimated using OLS method. Consequently, estimation of threshold values should be done first. The grid search method is used to find the potential threshold in the series by minimizing the residual sum of square of as follows:
argmin () Rss
, where is threshold parameter
The threshold grid search method only considers around 70% of observations against their residual sum square. The first and last values of arranged observations are excluded to achieve the minimum number of observations in each regime. The model which have the smallest residual sum of squares will have the most consistent estimate of the delay parameter. Therefore, a threshold value corresponding to the smallest sum square of residuals is efficient.
3.5.2.Multivariate time series model
3.5.2.1. Vector Autoregressive Model
The VAR model is one of the most successful, flexible, and easy to use models for the analysis of multivariate time series. It is a natural extension of the univariate autoregressive model to dynamic multivariate time series. The VAR model has proven to be especially useful for describing the dynamic behavior of economic and financial time series and for forecasting. Forecasts from VAR models are quite flexible because they can be made conditional on the potential future paths of specified variables in the model. A VAR system contains a set of m variables, each of which is expressed as a linear function of p lags of itself and of all of the other
m – 1 variables, plus an error term. A vector autoregressive order p, VAR(p) model can be
18 yt c1yt12yt2...pytpt
...
(15) Where t1,...,T and ytis a process equals with (y1t,y2t,..., ynt)' denote an (nx1) vector of time series variables, the variables yt could be in levels or in first differences, this depends on the nature of the data. i are (nxn) coefficient matrix and tis an (nx1) vector with zero mean white noise process.The VAR(p) can be written as in a lag operator form (L)Y Ct
Where (L)In1L...pLp.
The VAR(p) model is stable when the det(In1L...pLp)0, lie outside the complex unit circle.
VAR Order Selection
Like other time series models, the VAR model should also have the optimal lag length. The general approach is to fit VAR models with orders m = 0, ... , pmax and choose the value of m which minimizes some model selection criteria (Lutkepohl, 2005). The model selection criteria have a general form of
C(m)log|m|cT.(m,k) Where, t T t t m T ' 1 1
is the residual covariance matrix estimator for a model of order m,
) ,
(m k
is a function of order m which penalizes large VAR orders and cT is a sequence which
may depend on the sample size and identifies the specific criterion. The term log| m|
is a non-increasing function of the order m while (m,k) increases with m .The lag order is chosen which optimally balances these two forces.
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The three most commonly used information criteria for selecting the lag order are the Akaike information criterion (AIC), Schwarz information criterion (SC), and Hannan-Quin (HQ) information criteria: log| | 2mk2 T AIC m ( ) log| | log mk2 T T m SC m ( ) log| | log mk2 T T m HQ m
In each case (m,k)mk2 is the number of VAR parameters in a model with order m and k is number of variables. Denoting by p( AIC)
, p(SC)
and p(HQ)
the order selected by AIC, SC and HQ, respectively, the following relations hold for samples of fixed size T 16(Lutkepohl, 2005). p(SC) p(HQ) p(AIC)
Thus, among the three criteria AIC always suggests the largest order, SC chooses the smallest order and HQ is between. Of course, this does not preclude the possibility that all three criteria agree in their choice of VAR order. The HQ and SC criteria are both consistent, that is, the order estimated with these criteria converges in probability or almost surely to the true VAR order p under quit general conditions, if pmax exceeds the true order.
GRANGE CAUSALITY TEST
Grange causality is used to determine whether one-time series causes for another. It shows how useful one variable (or set of variables) x for forecasting another variable(s) y. When x Grange causes, y will be better predicted using the information of both x and y than its history alone. Thus, if x does not Grange cause y, it does not help to forecast y. The causality does not imply the change in one variable causes changes for another, it instead shows the correlation between the current value of a variable and the past value of another variable. The test could be x cause to
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Consider two variables xt and yt, the regression of yton lagged xtand lagged ytis
t j t m j i i t n i i t y x y
1 1Thenxt does not cause ytif, i 0,i1,..., p
Therefore, the test hypothesis for the Grange causality is:
H0:12 ...m 0
The test statistic of the Wald test (W) equals to J × F. Thereby, J is the number of restrictions to test (in the above case J = m). F denotes the value of the F statistic with
) /( ' / ) ' ' ( K T J F r r where r r
' equals the sum of the squared residuals by imposition of restrictions
) 0 ... (12 m and
' is the sum of squared residuals of the estimation without restrictions. T is the number of observations and K the number of regressors of the model. The test statistic W is asymptotically 2 distributed with J degrees of freedom. The rejection of the null is the sign of the causality of x to y. The test procedure for causality of y to x is the same.
COINTEGRATION
The long-run equilibrium relationship between variables in VAR system is known as the cointegrating vector. When there is a significant cointegrating vector, the VAR model should be augmented with an Error Correction term. In other words, pure VAR can be applied only when there is no cointegrating relationship among the variables in the VAR system. Hence, a prerequisite for running any VAR model is to run a cointegration test.
The role of cointegration is to link between the relations among a set of integrated nonstationary series and the long-term equilibrium. The presence of a cointegrating equation is interpreted as a long-run equilibrium relationship among the variables. If there is a set of k integrated variables of order one-I(1), there may exist up to k-1 independent linear relationships that are I(0). In general, there can be r ≤ k-1 linearly independent cointegrating vectors, which are gathered together into
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the k x r cointegrating matrix. Thus, each element in the r-dimensional vector is I(0), while each element in the k-dimensional vector is I(1) (Engle and Granger, 1987).
Testing for cointegration using Johansen’s
Let r be the rank of . Where is a matrix of vector of adjustment parameters and vectors of cointegrating parameters , '.
The maximum eigenvalue test and trace test are types of Johansen cointegration. This study used a trace cointegration test whether long-run equilibrium relationship between variables with the null hypothesis of no cointegration.
Trace Test
The trace statistics test the null hypothesis that there are at most r0 cointegrated relation and against the alternative r0 rank()n cointegrated relations. Where nis the maximum number of possible conitegrating vectors. If the null hypothesis is rejected, the number of cointgrating relations under the null becomes r01, rank r01 and the alternative r01rank()n. The testing procedure is the same as the test for the maximum eigenvalue test. The likelihood ratio test statistic is
LR(r0,n)Tln(1i)
Where LR(r0,n)is the likelihood ratio statistic for testing rank r versus the alternative
hypothesis that rank n
3.6 MODEL CHECKING
Model diagnostics is the important part of time series analysis, performed by the residual analysis. The residuals of the fitted model should have a white noise property where residuals are normally distributed with mean zero and constant variance, and have no autocorrelation problem.
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3.6.1 Residual Analysis
When residuals are not white noise the estimated variances of parameters becomes biased and inconsistent, also tests are invalid under model estimation. Besides, forecasts based on the model are inefficient due to high variance of forecast errors. Thus, performing residuals analysis before using models for particular purpose worth attention.
3.6.1.1 Residual autocorrelation test
The residuals autocorrelation analysis is performed using both graphical (informal) and statistical test. The one way of checking the autocorrelation structure of the residuals is to plot the autocorrelation and partial autocorrelation of residuals. The plots help to show if there is any autocorrelation in the residuals, suggesting that there is some information that has not been included in the model. Another way is plots of residuals versus their lags. Plot et horizontally and
et-1 vertically. i.e. plotting of the following observations (e1,e2 ), (e2,e3),…,(en,en+1 ). The falling
of points in quadrant one and three is a sign of residuals are positively autocorrelated. The failing of most of the points in quadrant two and four is a sign of negative autocorrelation. However, when the points are scattered equally in all quadrants residuals are random.
Univariate Portmanteau test of Residuals autocorrelation
The portmanteau test is used to check the autocorrelation structure of the residuals. The test hypothesis is:
H0: residuals are not serially correlated
H1:at least one successive residuals are serially correlated.
To test this null, Box and pierce (1970) suggested the Q-statistics.
K k k r n Q 1 2where k is the number of lag length, n is the number of observations and rk is the ACF of the residuals series at lag k. Under the null Q is asymptotically distributed k2pq. In a finite
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sample, the Q-statistic (17) may not be well approximated by k2pq. The modified Q-statistic suggested by Ljung and Box is:
K k k k n r n n Q 1 2 ~ ) 2 (Therefore, if the model is correct then, 2 ~
~ k p q
Q . The decision of the test is reject the null hypothesis at level of significance, if 2 (1 )
~
kpq
Q . Implying that the autocorrelation exists in residuals and assumption is violated.
Multivariate LaGrange Multiplier Test of Autocorrelation
Suppose a VAR model for the error ut given by
ut D1ut1...Dhuthvt
The vt denotes a white nose error term. Thus, to test autocorrelation in ut, the null hypothesis is H0:D1 ...Dh 0 against H1:Dj 0 for at least one jh
Using the Lagrange Multiplier test under the null hypothesis the study need to estimate the regular VAR model (ut= vt). To determine the test statistic the auxiliary model
UBZDUE where U (u1,...,uT) Zt [1TytT...ytTp1]T Z [Z0...ZT1] D[D1...Dh]
Define Fi such that
T t T i t t T iU u u F U 1 1
Then F [F1...Fh]24 U (I U)FT
This yields the least squares estimate of D
[ ( ) 1 ]1 T T T T T U Z ZZ Z U U U U U D
The standard 2 test statistic for testing whether D = 0 (no autocorrelation) is
Under H0 ( ) ( ) {[ ( ) ] } ( ) 1 vec D UU UZ ZZ ZU vec D h u T T T T T LM (h) 2(hk2) d LM
The decision of the test is reject the null hypothesis at level of significance, if ) 1 ( 2 2 hk
LM , implying that residuals are not random.
3.6.1.2 Residuals normality test
The assumption of normality must be satisfied for conventional tests of significance of coefficients and other statistics of the model to be valid. Normality residuals test will be performed using both graphical and statistical test.
Univariate Jarque-Bera test of normality of residuals
The most common test for normality of residuals is the Jarque-Bera test. The Jarque-Bera test assumes a variable being normally distributed with zero skewness and kurtosis equals to three. The null hypothesis of the test is:
H0: residuals are normally distributed
H1: residuals are not normally distributed
and the univariate Jarque-Bera test statistic is
4 ) 3 ( 6 2 2 kurtosis skewness n JB
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Under the hull hypothesis residuals are normally distributed
~ 2,2 Aprrox JB Therefore, if the 2 2 ,
JB , then reject the null hypothesis that residuals are normally distributed.
Multivariate Jarque-Bera test of normality of residuals
The multivariate Jarque-Berta test is used to test the multivariate normality of residuals in vector autoregressive model (ut). Like the univarate the test consider the skewness and kurtosis
properties of the ut (3rd & 4th moments) against those of a multivariate normal distribution of
the appropriate dimension. The test hypothesis is:
0 ) ( : 3 0 s t U E H and E(Uts)4 3 3 ) ( and 0 ) ( : 3 4 1 s t s t E U U E H
It is possible that the first four moments of the ut match the multivariate normal moments, and the ut is still not normally distributed. Formulation of the Jarque-Bera test uses a mean adjusted form of the VAR (p) model
( ) 1( 1 ) ... ( ) y y A y y A y y ut t p t p
T t T t t u u u kp T 1 1 1 Let pbe the matrix satisfying
u T p
p such that lim( )0
p p
p . Then the standardized residuals and their sample moments define as
t t p u w 1
T t it i k w T b b b b 1 3 1 1 11 1 1 ) ... ( and
T t it i k w T b b b b 1 4 2 2 12 2 1 ) ... (26 1 1/6 Tb b T s ( 231) ( 231)/24 b b T T k sk s k
The third and fourth moments of ut should be 0 and 3 respectively. Under the third moment assumption
2(k) d
s
Under the fourth moment assumption
2(k) d
k
Under both assumptions
2(k) d
sk
3.7 FORECASTING
Forecasting is the decision-making tool used at levels to help in budgeting, planning, and estimating future conditions. In the simplest terms, forecasting is the process of making predictions of the future based on past and present data and analysis of trends.
Forecasting performance method
Models should not be selected only based on their goodness of fit to the data but also based on the aim of the analysis. Since the objective of this study is to forecast the unemployment rate using out-of- sample forecasting method with the appropriate model, selection of models should perform carefully. However, sometimes the forecast ability of models which are best in the in-sample fitting may not provide more accurate results. Hence, to avoid this problem the study uses both the in-sample and out-of-sample forecasting performance methods. In general, empirical evidence from the out-of-sample forecasting performance is considered as more trustworthy than evidence from in-sample performance, which can be more sensitive to outliers and data mining. Consequently, a model with good performance in the out-of-sample forecasting performance is picked as the best model. The forecasting performance of models is done by
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sample data from data 1983q1 to 2010q4 used to models estimation and in-sample forecasting performance evaluation and sample data from 2011q1-2015q4 used to evaluate out-of-sample forecasting performance.
3.7.1 Out sample forecasting method
The two types of-sample forecasting methods are the direct of-sample and recursive out-of-sample. In practice, the recursive out-of-sample forecasting method gives accurate and unbiased forecasts. Hence, the study uses a recursive out-of-sample forecasting method.
Recursive out-sample forecasts
The recursive multistep forecasting method is a technique for predicting a sequence of values. It is a step by step prediction way, the predictive model is re-estimate at each step after the current predicted value is added to the data to predict the next one and the process will continue until the demanded values have predicted.
Suppose T r1,...,rT is total observation, which divided into (r1,...,rn) and (rn1,...,rT), where n is initial forecast origin, h is the forecast horizon. An initial sample data from t r1,...,rn is used to estimate each models, and h-step ahead out-of-sample forecasts are produced starting at time
n
r . The sample is increased by one, models are re-estimated, and h-step ahead forecasts are produced starting at rn 1. The iteration stops when the forecast origin is nT. In this way,
there will be (T nh) h-step ahead forecast errors for each model.
3.7.2 Forecasting Accuracy
Forecasting accuracy is a criterion to measure the forecasting performance of models. It measures the goodness fit of the forecasting model shows how able to predict the future values. The study compares the forecast ability of each model through the error statistics (criteria). Three error statistics are employed to measure the performance of models. Namely, the Root Mean Squared Error (RMSE), the Mean Absolute Error (MAE), and the Mean Absolute Percent Error (MAPE). The better forecast performance of the model is that with the smaller error statistics.
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Suppose yth/t is the h-step ahead forecast ofyth, their corresponding forecast error can be define as eth/t ythyth/t. Then, the forecast evaluation statistics based on N h-step ahead forecasts can be define as:
Root Mean Square Error
2 1 / 1
t N t j j h j e N RMSEMean Absolute Error
t N t j j h j e N MAE 1 / 1
Mean Absolute Percent Error
t N t j j h j h j y e N MAPE 1 / 1
The root mean square error (RMSE) is used to measure the difference between the predicted and actual values of the series. It is a measure of predictive power. The mean absolute error is the average absolute errors used to measure how the predicted values are close to the actual values. Besides, the mean absolute percent error measures the accuracy as the percentage of errors. Therefore, the better forecasting ability of the model is that with the smaller value of RMSE, MAE, and MAPE.
Diebold-Mariano (DM) test
A test suggested by Diebold and Mariano (1995) will be used to compare the forecasting performance of models. The test checks for the existence of significant differences between the forecasting accuracy of two models. The DM test has the null hypothesis of no difference between the forecast accuracy of the two models.
suppose yt is the series to be forecasted, y1th/ t and yt2h/ tis the two competing forecasts of yth
from two models i=1,2. The forecast errors from the two models are:
1 1 / /t tht h t ythy 2 2 / /t tht h t ythy
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The h-steps forecasts are computed for tt0,...,T for a total of T0 forecasts giving
T t T t tht t h t / 0 / 0 2 1 , Since the h-step forecasts use overlapping data the forecast errors in
T t t h t / 0 1 and
T t t h t / 0 2 will be serially correlated.The accuracy of each forecast is measured by particular loss function L(yth,ytih/t)L(tih/t) which is in most cases taken as the squared errors or Absolute errors. The Diebold-Mariano test with the null hypothesis of equal predictive accuracy between the models has the following loss function and test statistic:
Lossfunction d ( ) ( 2 / )
1 /
t L tht L th t The null of equal predictive accuracy is
H0 E
L((1)th/t)
E L((2)th/t)
The Diebold-Mariano test statistic is:2 / 1 2 / 1 ) / LRV ( )) ( avar ( T d d d S d Where
T t t t d T d 0 1 and 2 , cov( , ) 1 0 j t t j j j d d d LRV
dLRV is a constant estimate of the asymptotic (long run) variance of
d
T , it is also used in the test statistics because of the serially correlated sample of loss differentials for h>1. The DM test statistics S under the null of equal predictive accuracy approximates to a standard normal distributions S ~ N(0,1). Consequently, the null of equal predictive accuracy will reject at 5% level of significance if |S|>1.96.
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CHAPTER FOUR
4. RESULT AND DISCUSSION
The study used the quarterly time series data from January 1983 to December 2015. In this chapter, the results of the SARIMA(4,1,3)(0,0,1)4, SETAR(2,6,3) and VAR(5) models will
present. Model specification used for the forecasting of the unemployment rate. The discussion begins by describing the data set and the stationary test procedure. It followed by the modeling procedure and comparison of models performance. Finally, the unemployment rate will be forecasted with the selected model at different time horizon.
4.1. DESCRIPTIVE ANALYSIS
The descriptive statistics including the mean, standard deviation, coefficient of variation, minimum and maximum values of the economic variables under study are presented in table 1. The results show that the values of summary statistics are different especially industrial production has high mean and dispersion.
Table 1: Descriptive Statistics of Series:1983q1 to 2010q4
Variables Obs Mean Std.Dev Min Max
Unemployment rate 112 5.897321 2.671588 1.3 10.6 GDP percentage change of previous period 112 .5751832 1.006185 -3.709846 2.339088 Industrial production 112 84.03961 18.07617 55.11019 116.7097 Inflation 112 3.351751 3.153158 -1.41659 11.30091
4.2 STATIONARY TEST ANALYSIS
Before one attempts to fit a suitable time series model the data needs to test for a stationary property that is the series remains at a constant level over time. However, if a trend exists the series is not stationary. This examined graphically as well as using the Augmented Dickey-Fuller (ADF) and Phillips-Perron (PP) tests. The graph of economic variables at the level is presented in Figure 1. Except GDP percentage change of previous period the unemployment rate, industrial production and inflation are non-stationary at a level.
